Can two windows be active at the same time?
It is normal for different windows of a window manager to not get input (from the keyboard or mouse). That is, only one of them can be focused; and having the focus means that it gets keyboard and mouse input. Different applications can still get mouse and keyboard events however when they are aware of the other window: they can explicitly ask the window manager to still receive events. In the case of blender these aren't even two different applications; so I'm inclined to call it a bug of blender: blender should be aware where both windows are, and when the  mouse hovers over a split area of a window that doesn't have focus still process hot keys for that area. Blender could do the 'focus' part internally thus, and not care which of its windows has focus in the eyes of the window manager.